I want to reiterate from a previous post, that while I like the concept of the collective unconscious, I tend to think of it more like the waters of the earth - depending on a culture's worldview - sometimes the collective unconscious is sea, sometimes ocean, sometimes a freshwater lake or a tidal river - that these things are all water does not make them the "same". Each has its own unique ecological footprint and life-cycle. And so too, I think, with cultural archetypes.
I don't think one can assume total commonality and treat cross-cultural archetypes as variants of each other. They are embedded in inherently unique and vastly different worldviews. Culture matters. Context matters. Language Matters. Worldviews matter. To both the understanding of archetypes and the systems within which those archetypes arose and continue to have life.
Maybe the collective unconscious as a global psyche ecosystem could have, like the physical environment, a taxonomy of archetypes. Though there is a whole lot of risk to that kind of model because it lends itself too easily to hierarchy and racism and supremacy notions. Homo sapiens are one species and on the biological taxonomy everyone falls into the same categories. So taxonomy is not a clean fit.
Alternatively and probably more appropriate, the tales ecoystem of fairy-tales has the Aarne-Thompson classification system. The ATU index number for the tales contains tale types that have a common motif. For example, ATU 400–424 all feature brides or wives as the primary protagonist. This is the primary classification system for folklore/fairy tales and while it has drawbacks, (not sufficiently global - largely focused on North and West European tales, and that choosing a single motif i.e. "animal brides" may end up classifying stories together incorrectly from a thematic perspective). It is still an enormously useful tool.
Please note, I still need to do research on what if any classification systems have been attempted since Jung's initial 12 Archetypes. I'm hoping someone in the psychology field has done one. It would actually make archetypes easier for layfolks to work with independently. It might also avoid some of the more egregious interpretations that arise in the tarot world - where the archetypes end up as a mixed mag of mulch that are, across decks, only partially coherent.